Plotting the solutions to this gives me exactly what I want, but that's only because I have taken the correct value of H(infinity)=-0.88447 from previous texts. (In this case I have taken infinity=4.5, the solution converges by this point)

I need to be able to find the value of this unknown boundary condition myself.

How do I go about modifying my code to obtain this value of H(infinty) for myself using a shooting method? Is the shooting method in fact the best way of evaluating H(infinity)?